Key Stage 3 maths · Algebra

KS3 algebra: notation, expressions and formulae explained

Algebra is the language of Key Stage 3 maths. Children learn the shorthand for writing algebra, substitute numbers into expressions and formulae, simplify by collecting like terms, expand and factorise using brackets, multiply two brackets together and rearrange a formula to change its subject. Much of GCSE maths builds on these skills.

Algebraic notation and formulae, explained

Algebra begins with shorthand. The multiplication sign is left out, so a × b is written ab, and 3 × y is written 3y, which means y + y + y. Powers are written a² for a × a and a³ for a × a × a, and a²b means a × a × b. A division a ÷ b is written as a fraction, a/b, and coefficients are written as fractions rather than decimals, so children write ½x or x/2 rather than 0.5x. Brackets mean “do this part first”, so 3(a + b) is three lots of the whole of a + b. Most of the errors in later algebra come from not reading this shorthand correctly, so it deserves time.

Children also learn the vocabulary that separates different kinds of algebra. An expression such as 3x + 5 is a phrase with no equals sign, an equation such as 3x + 5 = 20 says two things are equal and can be solved, a formula such as A = lw links quantities, and an inequality such as x > 3 compares them. The parts of an expression separated by plus and minus signs are terms, and a factor is a number, letter or expression that divides into another exactly, so 3 and x are both factors of 3x. Substitution means replacing letters with numbers, which is how children use formulae from science too, such as speed = distance ÷ time or v = u + at.

Simplifying and manipulating expressions is about keeping them equivalent, meaning they give the same value whatever number is put in. Children collect like terms, so 4x + 3y − x + 2y becomes 3x + 5y, because only terms with exactly the same letters and powers can be combined. They multiply a single term over a bracket, so 3(x + 4) = 3x + 12. They go the other way too and take out common factors, writing 12x + 18 as 6(2x + 3). Finally they expand the product of two or more binomials, such as (x + 5)(x − 2), usually with a grid or by multiplying every term in the first bracket by every term in the second.

The last strand is using standard formulae and rearranging them to change the subject. If v = u + at, a child might need a on its own. They undo the formula step by step, in the way they would solve an equation: take away u from both sides to get v − u = at, then divide both sides by t, giving a = (v − u)/t. Checking with simple numbers is a good habit: if u = 2, a = 3 and t = 4 then v = 14, and the new formula gives (14 − 2)/4 = 3.

Words your child will hear

Term: A single part of an expression, such as 3x or −7, separated by + or − signs.
Coefficient: The number multiplying a letter. In 5x² the coefficient is 5.
Like terms: Terms with exactly the same letters and powers, such as 4x and 7x, which can be collected.
Expression, equation, formula: An expression has no equals sign, an equation can be solved, and a formula links real quantities such as area or speed.
Expand: Multiply out brackets, so 3(x + 4) becomes 3x + 12.
Factorise: Write an expression as a product by taking out a common factor, so 12x + 18 becomes 6(2x + 3).
Subject of a formula: The letter on its own on one side. In v = u + at, the subject is v.

What the National Curriculum says

In Key Stage 3, pupils should be taught to:

Use and interpret algebraic notation, including: ab in place of a × b; 3y in place of y + y + y and 3 × y; a² in place of a × a, a³ in place of a × a × a, a²b in place of a × a × b; a/b in place of a ÷ b; coefficients written as fractions rather than as decimals; and brackets
Substitute numerical values into formulae and expressions, including scientific formulae
Understand and use the concepts and vocabulary of expressions, equations, inequalities, terms and factors
Simplify and manipulate algebraic expressions to maintain equivalence by: collecting like terms; multiplying a single term over a bracket; taking out common factors; and expanding products of 2 or more binomials
Understand and use standard mathematical formulae; rearrange formulae to change the subject

Objectives from the National Curriculum in England, © Crown copyright, used under the Open Government Licence v3.0.

A worked example

Expand and simplify (x + 5)(x − 2).

Multiply every term in the first bracket by every term in the second. A grid helps keep track of the signs, and the minus sign belongs to the 2.
```
        x       −2
  x     x²      −2x
  +5    +5x     −10
```
Write out the four products: x × x = x², x × (−2) = −2x, 5 × x = 5x and 5 × (−2) = −10.
```
x² − 2x + 5x − 10
```
Collect the like terms. The two x terms combine: −2x + 5x = 3x.
```
x² + 3x − 10
```
Check by substituting a number into both versions. With x = 3: (3 + 5)(3 − 2) = 8 × 1 = 8, and 9 + 9 − 10 = 8.

(x + 5)(x − 2) = x² + 3x − 10

Common mistakes and how to help

Reading 3y as 3 and y written side by side, so y = 4 gives 34

In arithmetic, two digits next to each other make a bigger number, so 34 looks natural. In algebra, 3y means 3 × y.

What helps: Always write the multiplication sign when substituting: 3 × 4 = 12. Using brackets for the substituted number, 3(4), shows the multiplication clearly.

Collecting unlike terms: 2x + 3y = 5xy

It feels as though an answer ought to be a single term. But x and y stand for two different unknown numbers, so 2x and 3y are not like terms and cannot be combined.

What helps: Test with numbers. If x = 2 and y = 1, then 2x + 3y is 4 + 3 = 7 but 5xy is 5 × 2 × 1 = 10, so they can’t be the same. It is best to avoid “2 apples and 3 pears” explanations, which many maths teachers steer clear of, because the letters stand for numbers, not objects. The expression 2x + 3y is already as simple as it gets.

Squaring a bracket as if it were two separate terms: (x + 3)² = x² + 9

Children apply the power to each term and leave out the middle term. But (x + 3)² means (x + 3)(x + 3).

What helps: Write the bracket twice and expand it with a grid: x² + 3x + 3x + 9 = x² + 6x + 9. Test with x = 1: (1 + 3)² is 16 while 1 + 9 is only 10.

Substituting into x² as though the 2 multiplies: x = 3 gives 6

The small raised 2 is easy to mistake for a coefficient, and 3 × 2 is a more familiar sum. A related slip makes 2x² with x = 3 come out as 36 instead of 18, because the child squares 2 × 3 rather than just the 3.

What helps: Say “x squared” aloud and write it as x × x. For 2x², square first and then multiply: 3² = 9, then 2 × 9 = 18.

How to help at home

Ask about the numbers behind a formula. Everyday costs often follow a formula: a phone tariff is a fixed fee plus a price per gigabyte, and a taxi fare is a charge plus a rate per mile. Write the rule in letters and substitute a few numbers together.
Use the check-by-numbers habit. Whenever they simplify or expand, ask them to try one easy number in the first and last lines. If the two answers agree, the algebra is probably right. If not, the number points to where it went wrong.
Rearrange something familiar. Start with speed = distance ÷ time and ask for the formula for time, then for distance. Then try it with real figures from a journey so the rearranged versions mean something.
Talk through the notation aloud. Ask your child to read an expression like 2a²b in words: “two times a times a times b”. It sounds simple, but saying it slowly clears up a surprising number of errors.

Builds on

Year 6: Introduction to algebra

Leads on to

Key Stage 4: Factorising, indices and identities
Key Stage 4: Straight-line and quadratic graphs
Key Stage 4: Quadratics, simultaneous equations, inequalities
Key Stage 4: Special sequences and the nth term
Key Stage 4: Algebraic fractions, proof and functions
Key Stage 4: Exponential, trig and circle graphs
Key Stage 4: Quadratic formula, iteration, inequalities
Key Stage 4: Quadratic sequences and surd ratios